Transcript Graphs of the Cosecant and Secant Functions

Chapter 8: Trigonometric Functions And
Applications
8.1 Angles and Their Measures
8.2 Trigonometric Functions and Fundamental Identities
8.3 Evaluating Trigonometric Functions
8.4 Applications of Right Triangles
8.5 The Circular Functions
8.6 Graphs of the Sine and Cosine Functions
8.7 Graphs of the Other Circular Functions
8.8 Harmonic Motion
Copyright © 2007 Pearson Education, Inc.
Slide 8-2
8.7 Graphs of the Other Trigonometric
Functions
Graphs of the Cosecant and Secant Functions
• Cosecant values are reciprocals of the corresponding
sine values.
– If sin x = 1, the value of csc x is 1. Similarly, if sin x = –1,
then csc x = –1.
– When 0 < sin x < 1, then csc x > 1. Similarly,
if –1 < sin x < 0, then csc x < –1.
– When sin x approaches 0, the csc x gets larger. The
graph of y = csc x approaches the vertical line x = 0.
– In fact, the vertical asymptotes are the lines x = n.
Copyright © 2007 Pearson Education, Inc.
Slide 8-3
Precalculus 1 and 2
Agenda
SWBAT:
Write the COSECANT values
of ALL the reference angles
from 0 to 2π. (remember
cosecant is the INVERSE of
sine!)
• Sketch the cosecant graph
Sketch the cosecant
values along an X-Y
axis.
Repeat with Secant…!
Copyright © 2007 Pearson Education, Inc.
• Sketch the secant graph
• Recognize the
• HW: MCAS area and
volume review…
Slide 8-4
Precalculus 1 and 2
Agenda
SWBAT:
Write the COSECANT values
of ALL the reference angles
from 0 to 2π. (remember
cosecant is the INVERSE of
sine!)
Sketch the cosecant values
along an X-Y axis.
Repeat with Secant…!
Remember TAN=sine/cosine
• Sketch the cosecant graph
• Sketch the secant graph
• Recognize that we have an
exam tomorrow!!! You need
to know…
• HW: Study!
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Slide 8-5
Advanced Algebra All Stars
Agenda: May 6th, 2011
SWBAT:
Do Now – Last page in packet
– please hand in for credit!
• Welcome our guest 
Class work-IN PAIRS,
work on MCAS packet.
You will be challengedmeet the challenge!!!
• Work efficiently and
effectively with partner to
complete the MCAS
Challenge packet!!!
….if done….start
HW: Reflections Worksheet!
Copyright © 2007 Pearson Education, Inc.
• Tackle problems as a
TEAM!
Slide 8-6
Precalculus 2
Agenda
SWBAT:
Do Now – PEMDAS again.
Yeah, you should know it!
Write the COSECANT values
of ALL the reference angles
from 0 to 2π. (remember
cosecant is the INVERSE of
sine!)
Sketch the cosecant values
along an X-Y axis.
Repeat with Secant…!
Remember TAN=sine/cosine
• Identify transformations in
Sine and Cosine Graphs
Copyright © 2007 Pearson Education, Inc.
• Sketch the cosecant graph
• Sketch the secant graph
• Identify what COSECANT
and SECANT graphs look
like and explain WHY!
Slide 8-7
8.7 Graphs of the Cosecant and Secant
Functions
• A similar analysis for the secant function can be
done. Plotting a few points, we have the solid lines
representing the curves for the cosecant and secant
functions.
Copyright © 2007 Pearson Education, Inc.
Slide 8-8
8.7 Graphs of the Cosecant and Secant
Functions
• Cosecant Function
– Discontinuous at values of x of the form x = n, and has
vertical asymptotes at these values.
– No x-intercepts.
– Its period is 2 with no amplitude.
– Symmetric with respect to the origin, and is an odd
function.
• Secant Function
– Discontinuous at values of x of the form (2n + 1) 2 , and
has vertical asymptotes at these values.
– No x-intercepts.
– Its period is 2 with no amplitude.
– Symmetric with respect to the y-axis, and is an even
function.
Copyright © 2007 Pearson Education, Inc.
Slide 8-9
8.7 Sketching Traditional Graphs of the
Cosecant and Secant Functions
To graph y = a csc bx or y = a sec bx, with b > 0,
1. Graph the corresponding reciprocal function as a guide,
using a dashed curve.
To Graph
Use as a Guide
y = a csc bx
y = a sec bx
y = a sin bx
y = a cos bx
2. Sketch the vertical asymptotes. They will have equations
of the form x = k, k an x-intercept of the guide function.
3. Sketch the graph of the desired function by drawing the
U-shaped branches between adjacent asymptotes.
Copyright © 2007 Pearson Education, Inc.
Slide 8-10
8.7 Graphing y = a sec bx
1
Example Graph y  2 sec x.
2
1
Solution The guide function is y  2 cos x.
2
One period of the graph lies along the interval that
satisfies the inequality
1
0  x  2 , or [0,4 ].
2
Dividing this interval into four equal parts gives the
key points (0,2), (,0), (2,–2), (3,0), and (4,2),
which are joined with a smooth dashed curve.
Copyright © 2007 Pearson Education, Inc.
Slide 8-11
8.7
Graphing y = a sec bx
Sketch vertical asymptotes where the guide function
equals 0 and draw the U-shaped branches,
approaching the asymptotes.
Copyright © 2007 Pearson Education, Inc.
Slide 8-12
8.7 Graphs of Tangent and Cotangent
Functions
• Tangent
– Its period is  and it has no amplitude.
– Its values are 0 when sine values are 0, and undefined
when cosine values are 0.
– As x goes from  2 to 2 , tangent values go from – to
, and increase throughout the interval.
– The x-intercepts are of the form x = n.
Copyright © 2007 Pearson Education, Inc.
Slide 8-13
8.7 Graphs of Tangent and Cotangent
Functions
• Cotangent
– Its period is  and it has no amplitude.
– Its values are 0 when cosine values are 0, and undefined
when sine values are 0.
– As x goes from 0 to , cotangent values go from  to
–, and decrease throughout the interval.
– The x-intercepts are of the form x = (2n + 1) 2 .
Copyright © 2007 Pearson Education, Inc.
Slide 8-14
8.7 Sketching Traditional Graphs of the
Tangent and Cotangent Functions
To graph y = a tan bx or y = a cot bx, with b > 0,
1. The period is b . To locate two adjacent vertical
asymptotes, solve the following equations for x:

For y = a tan bx:
bx =  2
and
bx = 2 .
For y = a cot bx:
bx = 0
and
bx = .
2. Sketch the two vertical asymptotes found in Step 1.
3. Divide the interval formed by the vertical asymptotes into
four equal parts.
4. Evaluate the function for the first-quarter point, midpoint,
and third-quarter point, using x-values from Step 3.
5. Join the points with a smooth curve approaching the
vertical asymptotes.
Copyright © 2007 Pearson Education, Inc.
Slide 8-15
8.7 Graphing y = a cot bx
Example
1
Graph y  cot 2 x.
2
Solution
Since the function involves cotangent, we can
locate two adjacent asymptotes by solving the equations:
2 x  0  x  0.
2x    x 

2
.
Dividing the interval 0  x  2
into four equal parts and finding
the key points, we get
  , 1 ,   ,0 ,  3 , 1 .




 8 2  4   8 2
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Slide 8-16